Topological chaos, braiding and bifurcation of almost-cyclic sets

TL;DR

This paper explores how the braiding of almost-cyclic sets in 2D flows can analyze chaos, showing that topological entropy bounds derived from braids effectively characterize flow complexity.

Contribution

It extends the application of the Thurston-Nielsen classification to almost-cyclic sets, linking topological braiding with chaos analysis in fluid flows.

Findings

Braids from almost-cyclic sets provide good lower bounds on topological entropy.

Bifurcations of almost-cyclic sets lead to topologically distinct braids.

Topological analysis can be integrated with statistical methods for complex systems.

Abstract

In certain two-dimensional time-dependent flows, the braiding of periodic orbits provides a way to analyze chaos in the system through application of the Thurston-Nielsen classification theorem (TNCT). We expand upon earlier work that introduced the application of the TNCT to braiding of almost-cyclic sets, which are individual components of almost-invariant sets [Stremler, Ross, Grover, Kumar, Topological chaos and periodic braiding of almost-cyclic sets. Physical Review Letters 106 (2011), 114101]. In this context, almost-cyclic sets are periodic regions in the flow with high local residence time that act as stirrers or `ghost rods' around which the surrounding fluid appears to be stretched and folded. In the present work, we discuss the bifurcation of the almost-cyclic sets as a system parameter is varied, which results in a sequence of topologically distinct braids. We show that, for Stokes' flow in a lid-driven cavity, these various braids give good lower bounds on the topological entropy over the respective parameter regimes in which they exist. We make the case that a topological analysis based on spatiotemporal braiding of almost-cyclic sets can be used for analyzing chaos in fluid flows. Hence we further develop a connection between set-oriented statistical methods and topological methods, which promises to be an important analysis tool in the study of complex systems.